I've owned an Axe-FX Ultra from almost the beginning and still own it, but don't use it that much. I currently own and use the Kemper. I have written an extensive document on my experiences with these devices and my experiences with using them as well as interfacing with the respective companies. Maybe one day I'll post it here. But suffice it to say that the Kemper to *me* is the better product.
In a nutshell, the Kemper is far easier to use, tweak, and dial-in a tone that I like. The thousands of profiles out there are amazing, and the list continues to grow. Only after doing my own profiles of my own gear that I started to realize the power of the profiling concept, and the ability to share these profiles. In my opinion, this far exceeds any pseudo Ultra-Res IR concept (Google this and Jay Mitchell to see how bogus this concept really is). The Axe-FX simulates the circuit schematic and thus needs a schematic to start with. This is lower-level simulation (simulating the components such as tubes, transformers, etc.). Most every amp sim (HW or SW) does it this way. I love the Kemper philosophy of treating the amp like a black box and determining its distortion characteristics. This is simulation at the highest level and is truly innovative and novel.
Sure, the Axe-FX has an over-priced, dedicated foot controller, has more flexible routing, "better" effects, and a USB audio interface. But for Live Sound, I challenge anybody in the audience to hear the difference in effects using the same PA. And in recording -- true recordings use post-processing for those most of the time. Also, do I really need 4 parallel chains in stereo of effects and amps? Sure -- nice to have, but rarely used in most cases. Flexibility overkill.
Then there's the whole business side and attitudes of the Axe-FX team. It just doesn't sit well with me. The arrogance, condescension to customers, and poor behaviour of the people involved -- both from the company and their fan boys -- is astounding. I also don't buy all the gimmicks (Ultra-Res) IR and marketing tricks (I'm all sold out of the II -- but the II XL (my most expensive product) is available!).
If you want to just sit back and enjoy playing with awesome tone, with minimal amount of tweaking, as well as enjoy thousands of different amps, pedals, distortion devices, etc. -- then the Kemper is for you!